How to add an object with texture inside another object?
The first thing you need is to model a proper eye. Like so:


The outer part is 0.1cm away from the inner in this case.


The outer part needs to be refractive (or even just transparent) and it will also need specular as it will catch the high spec of the eye.

Depending on the render budget you have you can add SSS to the inner part. I did in this case but the difference here is subtle. In this case i'm aiming cartoon semirealistic eye but the principles are pretty much the same for any eye.

You now have to UV unwrap the inner part. Easy, divide the eye in two parts on its XZ plane center and Unwrap.



Now you can add the diffuse texture to the inner part of the eye, but note that for a proper eye render you will need more than just a diffuse shader.
Personally I add a specular map only to the iris using an high IOR and roughness value, a bump (or displacement if needed) map aswell.

Here is the final graph of my setup:


and a little GIF preview: